I wanted to post this as a new question to make sure you saw it. Thanks! :)

Posted by klynn on Wednesday, October 3, 2007 at 11:29am.

This is an MBA-level Managerial Economics Course. I'm working on some HW and just want to double-check my answers.

1. Jimbo's is a new company producing exploding cigars. Jimbo's company has the following demand curve for the cigars:

P = 10 - 2Q

Jimbo is currently charging $2 per cigar.

A. A marketing official cliams that the price elasticity of demand at $2 is -2.0. Do you agree? If you disagree, what is the correct price elasticity?

On A, I found the price elasticity to be -.25. In the problem, P=2 and Q=4 (as a result of P=2). With a 1% increase in P, the new P is $2.02. The new Q as a result of the price increase is 3.99. So, a $.02 increase in price led to a .01 decrease in Q demanded. Price elasticity is the % change in Q/% change in P, so:

-.01/4 divided by .02/2 =
-.0025/.01 =
-.25

Can someone tell me if this is correct? If not, can you tell me how to get the correct answer? Thanks! :)

For Further Reading

Managerial Economics/Math - economyst, Wednesday, October 3, 2007 at 2:33pm
I agree with your methodology and answer.

Managerial Economics/Math - klynn, Wednesday, October 3, 2007 at 4:11pm
Ok, great! Thanks! If you don't mind, the question does have a couple other steps that I need a little bit of help on.

B. Evaluate the wisdom of the pricing policy. If a price change is advisable, what price would you recommend that, with certainty, would improve profits the most and why? Given that you do not know marginal costs, state the necessary conditions for further price changes, up or down, from the price that you feel is certain to improve profits the most. Hint: think of the changing price elasticities as you slide down a linear demand curve.

For this one, I know that the top part of the demand curve is elastic, the middle is unit elasticity, and the bottom part is inelastic. Given that the elasticity is -.25, this (i think?) should fall into inelasticity. In the inelastic part of the demand curve, prices will rise. But, I'm not sure how to figure out exactly what price to charg since we don't have costs.

C. If, in fact, Jimbo's company has a demand curve equal to Q = 100P^-2 and constant marginal costs of $4 per cigar, what would be the profit maximizing price for the cigars?

On this one, I know I need to set MC = MR to find the profit maximizing point. MR = 200P^-3 and MC =4. So, this is 4 = 200P^-3, which then goes down to -50 = P^-3 (i think?). But, I'm not sure how to get P by itself b/c the -3 exponent is throwing me off.

Any help is much appreciated. Thanks so much for all your help! Without you, I'm not sure how well I'd be making it through this class! :)

For B) First, if the firm faces a downward sloping demand curve, then the firm has some monopoly power -- Use the general monopoly model to figure your solutions. While true that you don't know what costs are, you can be certain, at least, that marginal costs are positive. So, the firm can certainly do better if it raises price such that MR=MC=0. BTW, at MC=0 with the given demand curve, you would find the price elasticity to be -1.0.

For C) I see you have made a math error, which has gotten you off track.
For these kinds of problem, I (personally) like to see demand equations of the form P=f(Q) (Like the equation for Jimbo cigars above. This way, I don't get confused about what the demand equation would look like in a graph). So, do some simple algebra and re-arrange terms.
You start with Q=100/(P^2). So, P^2 = 100/Q. So P=sqrt(100/Q) = 10/(Q^.5)

Now then TR=P*Q = 10*Q^.5.
MR is the first derivative. MR=5/(Q^.5)
Take it from here.

To answer Part A of the question, you correctly calculated the price elasticity of demand as -0.25. This means that for a 1% increase in price, the quantity demanded will decrease by 0.25%.

To calculate price elasticity of demand, you can use the formula:
Price elasticity of demand = (% change in quantity demanded) / (% change in price)

In your calculation, you used the values P=2 and Q=4. Then you increased the price by 1% to $2.02, which resulted in a new quantity demanded of 3.99.

On the numerator, you calculated the percentage change in quantity demanded as 0.01 (which is the absolute value of the change, not negative). On the denominator, you calculated the percentage change in price as 0.02 (1% of $2).

Dividing the numerator by the denominator, you correctly found the price elasticity of demand as -0.25.

For Part B of the question, you correctly identified that the price elasticity of demand is -0.25, which falls into the inelastic range. In the inelastic part of the demand curve, prices will rise.

However, since you do not have information about the marginal costs, it becomes challenging to determine the optimal price that would improve profits the most. In general, to determine the best price, you need to consider both the price elasticity of demand and the cost structure of the business. Without knowing the marginal costs, it is not possible to recommend a specific price.

The necessary conditions for further price changes, either up or down, will depend on the specific cost structure and elasticity of demand. Generally, if the demand is inelastic, price increases could potentially increase profits, as long as the increased revenue outweighs any potential decrease in demand. Conversely, if the demand is elastic, price decreases might be more suitable to increase profits, as the increase in quantity demanded from lower prices could offset any decrease in revenue per unit.

For Part C of the question, you are given the demand curve Q = 100P^-2 and constant marginal costs of $4 per cigar. To find the profit-maximizing price, you need to set marginal cost (MC) equal to marginal revenue (MR).

In this case, marginal revenue is given by MR = 200P^-3 (derived from differentiating Q with respect to P).

Setting MR equal to MC:

200P^-3 = 4

To solve for P, you can rearrange the equation:

P^-3 = 4/200
P^-3 = 1/50

To get rid of the negative exponent, take the reciprocal of both sides:

1/P^3 = 50

Raising both sides to the power of -1/3:

P^3 = (1/50)^(-1/3)
P^3 = 50^(-1/3)

Taking the cube root of both sides:

P = 50^(-1/9)

This is the profit-maximizing price for the cigars.